By Daniel Liberzon

This textbook bargains a concise but rigorous creation to calculus of adaptations and optimum regulate thought, and is a self-contained source for graduate scholars in engineering, utilized arithmetic, and comparable matters. Designed in particular for a one-semester direction, the booklet starts off with calculus of adaptations, getting ready the floor for optimum keep watch over. It then provides a whole evidence of the utmost precept and covers key issues equivalent to the Hamilton-Jacobi-Bellman thought of dynamic programming and linear-quadratic optimum regulate.

*Calculus of diversifications and optimum regulate Theory* additionally strains the ancient improvement of the topic and lines quite a few workouts, notes and references on the finish of every bankruptcy, and recommendations for additional study.

- Offers a concise but rigorous creation
- Requires constrained heritage on top of things conception or complicated arithmetic
- Provides a whole evidence of the utmost precept
- Uses constant notation within the exposition of classical and glossy issues
- Traces the ancient improvement of the topic
- Solutions guide (available purely to teachers)

Leading universities that experience followed this ebook include:

- University of Illinois at Urbana-Champaign ECE 553: optimal keep an eye on structures
- Georgia Institute of expertise ECE 6553: optimum keep an eye on and Optimization
- collage of Pennsylvania ESE 680: optimum keep watch over Theory
- college of Notre Dame EE 60565: optimum Control

**Read or Download Calculus of Variations and Optimal Control Theory: A Concise Introduction PDF**

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**Extra resources for Calculus of Variations and Optimal Control Theory: A Concise Introduction**

**Example text**

11) In other words, we must only consider perturbations vanishing at the endpoints. 11) we must have δJ|y (η) = 0. 9) to arrive at a more explicit condition in terms of the Lagrangian L. Recall that the first variation δJ|y was defined via J(y + αη) = J(y) + δJ|y (η)α + o(α). 12) is b L(x, y(x) + αη(x), y (x) + αη (x))dx. 13) a We can write down its first-order Taylor expansion with respect to α by expanding the expression inside the integral with the help of the chain rule: b L(x, y(x), y (x)) + Ly (x, y(x), y (x))αη(x) J(y + αη) = a + Lz (x, y(x), y (x))αη (x) + o(α) dx.

It might not be necessary to actually perform all of these operations, though, as the next example demonstrates. 2 Let us find the shortest path between two points in the plane. Of course the answer is obvious, but let us see how we can obtain it from the Euler-Lagrange equation. 1, in the context of Dido’s problem and the catenary problem (where it played the role of a side constraint rather than b the cost functional). This length functional is J(y) = a 1 + (y (x))2 dx, √ hence the Lagrangian is L(x, y, z) = 1 + z 2 .

M. 0= dα α=0 We have shown that for an arbitrary C 1 curve x(·) in D with x(0) = x∗ , its tangent vector x (0) must satisfy ∇hi (x∗ ) · x (0) = 0 for each i. Actually, one can show that the converse is also true, namely, every vector d ∈ Rn satisfying ∇hi (x∗ ) · d = 0, i = 1, . . 20) is a tangent vector to D at x∗ corresponding to some curve. 20) holds. This is the characterization of the tangent space T x∗ D that we were looking for. 20) that Tx∗ D is a subspace of Rn ; in particular, if d is a tangent vector, then so is −d (going from x (0) to −x (0) corresponds to reversing the direction of the curve).